Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 306 / 2 / d

Take $D$ embedding coordinates and give one coordinate the background charge $q$. The remaining $D-1$ free bosons contribute $D-1$, while the distinguished coordinate contributes $1+6{\alpha }^{\prime }{q}^2$, so the matter central charge isThe worldsheet ghosts contributes $-26$. Cancellation of the worldsheet Weyl anomaly therefore requireswhich is real for $D<26$ and produces a noncritical bosonic string.

In target-space language the dilaton is linear in this coordinate, so the local string coupling $g_s=e^{\Phi }$ changes exponentially. One end of the target direction is weakly coupled and the other is strongly coupled. Consequently string perturbation theory is trustworthy only in the weak-coupling region, rather than throughout the full background.